This article gives a starting point idea of identities. Here we will define the identity, introduce the notation used, and, of course, give various examples identities
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What is identity?
It is logical to start presenting the material with identity definitions. In Makarychev Yu. N.’s textbook, algebra for 7th grade, the definition of identity is given as follows:
Definition.
Identity– this is an equality that is true for any values of the variables; any true numerical equality is also an identity.
At the same time, the author immediately stipulates that in the future this definition will be clarified. This clarification occurs in 8th grade, after becoming familiar with the definition of permissible values of variables and DL. The definition becomes:
Definition.
Identities- these are true numerical equalities, as well as equalities that are true for all acceptable values the variables included in them.
So why, when defining identity, in 7th grade we talk about any values of variables, and in 8th grade we start talking about the values of variables from their DL? Until grade 8, work is carried out exclusively with whole expressions (in particular, with monomials and polynomials), and they make sense for any values of the variables included in them. That’s why in 7th grade we say that identity is an equality that is true for any values of the variables. And in the 8th grade, expressions appear that no longer make sense not for all values of variables, but only for values from their ODZ. Therefore, we begin to call equalities that are true for all admissible values of the variables.
So identity is special case equality. That is, any identity is equality. But not every equality is an identity, but only an equality that is true for any values of the variables from their range of permissible values.
Identity sign
It is known that in writing equalities, an equal sign of the form “=” is used, to the left and to the right of which there are some numbers or expressions. If we add another horizontal line to this sign, we get identity sign“≡”, or as it is also called equal sign.
The sign of identity is usually used only when it is necessary to especially emphasize that we are faced with not just equality, but identity. In other cases, records of identities do not differ in appearance from equalities.
Examples of identities
It's time to bring examples of identities. The definition of identity given in the first paragraph will help us with this.
Numerical equalities 2=2 are examples of identities, since these equalities are true, and any true numerical equality is by definition an identity. They can be written as 2≡2 and .
Numerical equalities of the form 2+3=5 and 7−1=2·3 are also identities, since these equalities are true. That is, 2+3≡5 and 7−1≡2·3.
Let's move on to examples of identities that contain not only numbers, but also variables.
Consider the equality 3·(x+1)=3·x+3. For any value of the variable x, the written equality is true due to the distributive property of multiplication relative to addition, therefore, the original equality is an example of an identity. Here is another example of an identity: y·(x−1)≡(x−1)·x:x·y 2:y, here the range of permissible values of the variables x and y consists of all pairs (x, y), where x and y are any numbers except zero.
But the equalities x+1=x−1 and a+2·b=b+2·a are not identities, since there are values of the variables for which these equalities will not be true. For example, when x=2, the equality x+1=x−1 turns into the incorrect equality 2+1=2−1. Moreover, the equality x+1=x−1 is not achieved at all for any values of the variable x. And the equality a+2·b=b+2·a will turn into an incorrect equality if we take any different values of the variables a and b. For example, with a=0 and b=1 we will arrive at the incorrect equality 0+2·1=1+2·0. Equality |x|=x, where |x| - variable x is also not an identity, since it is not true for negative values x.
Examples of the most well-known identities are of the form sin 2 α+cos 2 α=1 and a log a b =b.
In conclusion of this article, I would like to note that when studying mathematics we constantly encounter identities. Records of properties of actions with numbers are identities, for example, a+b=b+a, 1·a=a, 0·a=0 and a+(−a)=0. Also the identities are
Basic properties of addition and multiplication of numbers.
Commutative property of addition: rearranging the terms does not change the value of the sum. For any numbers a and b the equality is true
Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third to the first number. For any numbers a, b and c the equality is true
Commutative property of multiplication: rearranging the factors does not change the value of the product. For any numbers a, b and c the equality is true
Combinative property of multiplication: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third.
For any numbers a, b and c the equality is true
Distributive Property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true
From the commutative and combinative properties of addition it follows: in any sum you can rearrange the terms in any way you like and arbitrarily combine them into groups.
Example 1 Let's calculate the sum 1.23+13.5+4.27.
To do this, it is convenient to combine the first term with the third. We get:
1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.
From the commutative and combinative properties of multiplication it follows: in any product you can rearrange the factors in any way and arbitrarily combine them into groups.
Example 2 Let's find the value of the product 1.8·0.25·64·0.5.
Combining the first factor with the fourth, and the second with the third, we have:
1.8·0.25·64·0.5=(1.8·0.5)·(0.25·64)=0.9·16=14.4.
The distributive property is also true when a number is multiplied by the sum of three or more terms.
For example, for any numbers a, b, c and d the equality is true
a(b+c+d)=ab+ac+ad.
We know that subtraction can be replaced by addition by adding to the minuend the opposite number of the subtrahend:
This allows a numeric expression type a-b be considered the sum of numbers a and -b, a numerical expression of the form a+b-c-d be considered the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.
Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.
This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the properties of addition, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -4.
Example 4 Let's calculate the product 36·().
The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we obtain:
36()=36·-36·=9-10=-1.
Identities
Definition. Two expressions whose corresponding values are equal for any values of the variables are called identically equal.
Definition. An equality that is true for any values of the variables is called an identity.
Let's find the values of the expressions 3(x+y) and 3x+3y for x=5, y=4:
3(x+y)=3(5+4)=3 9=27,
3x+3y=3·5+3·4=15+12=27.
We got the same result. From the distribution property it follows that, in general, for any values of the variables, the corresponding values of the expressions 3(x+y) and 3x+3y are equal.
Let us now consider the expressions 2x+y and 2xy. When x=1, y=2 they take equal values:
However, you can specify values of x and y such that the values of these expressions are not equal. For example, if x=3, y=4, then
The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.
The equality 3(x+y)=x+3y, true for any values of x and y, is an identity.
True numerical equalities are also considered identities.
Thus, identities are equalities that express the basic properties of operations on numbers:
a+b=b+a, (a+b)+c=a+(b+c),
ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.
Other examples of identities can be given:
a+0=a, a+(-a)=0, a-b=a+(-b),
a·1=a, a·(-b)=-ab, (-a)(-b)=ab.
Identical transformations of expressions
Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression.
Identity transformations expressions with variables are performed based on the properties of operations on numbers.
To find the value of the expression xy-xz for given values of x, y, z, you need to perform three steps. For example, with x=2.3, y=0.8, z=0.2 we get:
xy-xz=2.3·0.8-2.3·0.2=1.84-0.46=1.38.
This result can be obtained by performing only two steps, if you use the expression x(y-z), which is identically equal to the expression xy-xz:
xy-xz=2.3(0.8-0.2)=2.3·0.6=1.38.
We have simplified the calculations by replacing the expression xy-xz with the identically equal expression x(y-z).
Identical transformations of expressions are widely used in calculating the values of expressions and solving other problems. Some identical transformations have already had to be performed, for example, bringing similar terms, opening parentheses. Let us recall the rules for performing these transformations:
to bring similar terms, you need to add their coefficients and multiply the result by the common letter part;
if there is a plus sign before the brackets, then the brackets can be omitted, preserving the sign of each term enclosed in brackets;
If there is a minus sign before the parentheses, then the parentheses can be omitted by changing the sign of each term enclosed in the parentheses.
Example 1 Let us present similar terms in the sum 5x+2x-3x.
Let's use the rule for reducing similar terms:
5x+2x-3x=(5+2-3)x=4x.
This transformation is based on the distributive property of multiplication.
Example 2 Let's open the brackets in the expression 2a+(b-3c).
Using the rule for opening parentheses preceded by a plus sign:
2a+(b-3c)=2a+b-3c.
The transformation carried out is based on the combinatory property of addition.
Example 3 Let's open the brackets in the expression a-(4b-c).
Let's use the rule for opening parentheses preceded by a minus sign:
a-(4b-c)=a-4b+c.
The transformation performed is based on the distributive property of multiplication and the combinatory property of addition. Let's show it. Let us represent the second term -(4b-c) in this expression as a product (-1)(4b-c):
a-(4b-c)=a+(-1)(4b-c).
Applying the specified properties of actions, we obtain:
a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.
After we have dealt with the concept of identities, we can move on to studying identically equal expressions. The purpose of this article is to explain what it is and show with examples which expressions will be identically equal to others.
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Identically equal expressions: definition
The concept of identically equal expressions is usually studied together with the concept of identity itself within the framework school course algebra. Here is the basic definition taken from one textbook:
Definition 1
Identically equal there will be expressions to each other whose values will be the same for any possible values variables included in their composition.
Also, those numerical expressions to which the same values will correspond are considered identically equal.
This is a fairly broad definition that will be true for all integer expressions whose meaning does not change when the values of the variables change. However, later there is a need to clarify this definition, since there are other types of expressions besides integers that will not make sense given certain variables. This gives rise to the concept of admissibility and inadmissibility of certain variable values, as well as the need to determine the range of permissible values. Let us formulate a refined definition.
Definition 2
Identically equal expressions– these are those expressions whose values are equal to each other for any permissible values of the variables included in their composition. Numerical expressions will be identically equal to each other provided the values are the same.
The phrase “for any valid values of the variables” indicates all those values of the variables for which both expressions will make sense. We will explain this point later when we give examples of identically equal expressions.
You can also provide the following definition:
Definition 3
Identically in equal terms expressions located in the same identity on the left and right sides are called.
Examples of expressions that are identically equal to each other
Using the definitions given above, let's look at a few examples of such expressions.
Let's start with numerical expressions.
Example 1
Thus, 2 + 4 and 4 + 2 will be identically equal to each other, since their results will be equal (6 and 6).
Example 2
In the same way, the expressions 3 and 30 are identically equal: 10, (2 2) 3 and 2 6 (to calculate the value of the last expression you need to know the properties of the degree).
Example 3
But the expressions 4 - 2 and 9 - 1 will not be equal, since their values are different.
Let's move on to examples literal expressions. a + b and b + a will be identically equal, and this does not depend on the values of the variables (the equality of expressions in this case is determined by the commutative property of addition).
Example 4
For example, if a is equal to 4 and b is equal to 5, then the results will still be the same.
Another example of identically equal expressions with letters is 0 · x · y · z and 0 . Whatever the values of the variables in this case, when multiplied by 0, they will give 0. The unequal expressions are 6 · x and 8 · x, since they will not be equal for any x.
In the event that the areas of permissible values of the variables coincide, for example, in the expressions a + 6 and 6 + a or a · b · 0 and 0, or x 4 and x, and the values of the expressions themselves are equal for any variables, then such expressions are considered identically equal. So, a + 8 = 8 + a for any value of a, and a · b · 0 = 0 too, since multiplying any number by 0 results in 0. The expressions x 4 and x will be identically equal for any x from the interval [ 0 , + ∞) .
But the range of valid values in one expression may be different from the range of another.
Example 5
For example, let's take two expressions: x − 1 and x - 1 · x x. For the first of them, the range of permissible values of x will be the entire set of real numbers, and for the second - the set of all real numbers, with the exception of zero, because then we will get 0 in the denominator, and such a division is not defined. These two expressions have a common range of values formed by the intersection of two separate ranges. We can conclude that both expressions x - 1 · x x and x − 1 will make sense for any real values of the variables, with the exception of 0.
The basic property of the fraction also allows us to conclude that x - 1 · x x and x − 1 will be equal for any x that is not 0. This means that on the general range of permissible values these expressions will be identically equal to each other, but for any real x we cannot speak of identical equality.
If we replace one expression with another, which is identically equal to it, then this process is called an identity transformation. This concept is very important, and we will talk about it in detail in a separate material.
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While studying algebra, we came across the concepts of a polynomial (for example ($y-x$,$\ 2x^2-2x$, etc.) and algebraic fraction (for example $\frac(x+5)(x)$, $\frac(2x ^2)(2x^2-2x)$,$\ \frac(x-y)(y-x)$, etc.) The similarity of these concepts is that both polynomials and algebraic fractions contain variables and numeric values, arithmetic operations are performed: addition, subtraction, multiplication, exponentiation. The difference between these concepts is that in polynomials division by a variable is not performed, but in algebraic fractions division by a variable can be performed.
Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are whole rational expressions, and algebraic fractions fractional-rational expressions.
Can be obtained from fractionally --rational expression whole algebraic expression using an identity transformation, which in this case will be the main property of a fraction - reduction of fractions. Let's check this in practice:
Example 1
Convert:$\ \frac(x^2-4x+4)(x-2)$
Solution: Convert given fractional rational equation is possible by using the basic property of the reduction fraction, i.e. dividing the numerator and denominator by the same number or expression other than $0$.
This fraction cannot be reduced immediately; the numerator must be converted.
Let's transform the expression in the numerator of the fraction, for this we use the formula for the square of the difference: $a^2-2ab+b^2=((a-b))^2$
The fraction looks like
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)\]
Now we see that there is a common factor in the numerator and denominator - this is the expression $x-2$, by which we will reduce the fraction
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)=x-2\]
After reduction, we found that the original fractional rational expression $\frac(x^2-4x+4)(x-2)$ became a polynomial $x-2$, i.e. whole rational.
Now let us pay attention to the fact that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2\ $ can be considered identical not for all values of the variable, because In order for a fractional rational expression to exist and to be able to reduce by the polynomial $x-2$, the denominator of the fraction must not be equal to $0$ (as well as the factor by which we are reducing. In in this example the denominator and the multiplier are the same, but this is not always the case).
The values of the variable at which the algebraic fraction will exist are called the permissible values of the variable.
Let's put a condition on the denominator of the fraction: $x-2≠0$, then $x≠2$.
This means that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2$ are identical for all values of the variable except $2$.
Definition 1
Identically equal expressions are those that are equal for all valid values of the variable.
An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include performing actions: addition, subtraction, multiplication, placing a common factor out of brackets, reduction algebraic fractions to a common denominator, reducing algebraic fractions, reducing similar terms, etc. It is necessary to take into account that a number of transformations, such as reduction, reduction of similar terms, can change the permissible values of the variable.
Techniques used to prove identities
Bring the left side of the identity to the right or vice versa using identity transformations
Reduce both sides to the same expression using identical transformations
Transfer the expressions in one part of the expression to another and prove that the resulting difference is equal to $0$
Which of the above techniques to use to prove a given identity depends on the original identity.
Example 2
Prove the identity $((a+b+c))^2- 2(ab+ac+bc)=a^2+b^2+c^2$
Solution: To prove this identity, we use the first of the above methods, namely, we will transform the left side of the identity until it is equal to the right.
Let's consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$ - it represents the difference of two polynomials. In this case, the first polynomial is the square of the sum of three terms. To square the sum of several terms, we use the formula:
\[((a+b+c))^2=a^2+b^2+c^2+2ab+2ac+2bc\]
To do this, we need to multiply a number by a polynomial. Remember that to do this we need to multiply the common factor behind the brackets by each term of the polynomial in the brackets. Then we get:
$2(ab+ac+bc)=2ab+2ac+2bc$
Now let's return to the original polynomial, it will take the form:
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)$
Please note that before the bracket there is a “-” sign, which means that when the brackets are opened, all the signs that were in the brackets change to the opposite.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc$
Let us present similar terms, then we obtain that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is $0$.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc=a^2+b^2+c^2$
This means that by means of identical transformations we have obtained an identical expression on the left side of the original identity
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2$
Note that the resulting expression shows that the original identity is true.
Please note that in the original identity all values of the variable are allowed, which means we proved the identity using identity transformations, and it is true for all possible values of the variable.
Let's consider two equalities:
1. a 12 *a 3 = a 7 *a 8
This equality will hold for any values of the variable a. The range of acceptable values for that equality will be the entire set of real numbers.
2. a 12: a 3 = a 2 *a 7 .
This inequality will be true for all values of the variable a, except for a equal to zero. The range of acceptable values for this inequality will be the entire set of real numbers except zero.
For each of these equalities it can be argued that it will be true for any admissible values of the variables a. Such equalities in mathematics are called identities.
The concept of identity
An identity is an equality that is true for any admissible values of the variables. If you substitute any valid values into this equality instead of variables, you should get a correct numerical equality.
It is worth noting that true numerical equalities are also identities. Identities, for example, will be properties of actions on numbers.
3. a + b = b + a;
4. a + (b + c) = (a + b) + c;
6. a*(b*c) = (a*b)*c;
7. a*(b + c) = a*b + a*c;
11. a*(-1) = -a.
If two expressions for any admissible variables are respectively equal, then such expressions are called identically equal. Below are some examples of identically equal expressions:
1. (a 2) 4 and a 8 ;
2. a*b*(-a^2*b) and -a 3 *b 2 ;
3. ((x 3 *x 8)/x) and x 10.
We can always replace one expression with any other expression identically equal to the first. Such a replacement will be an identity transformation.
Examples of identities
Example 1: are the following equalities identical:
1. a + 5 = 5 + a;
2. a*(-b) = -a*b;
3. 3*a*3*b = 9*a*b;
Not all expressions presented above will be identities. Of these equalities, only 1, 2 and 3 equalities are identities. No matter what numbers we substitute in them, instead of variables a and b we will still get correct numerical equalities.
But 4 equality is no longer an identity. Because this equality will not hold for all valid values. For example, with the values a = 5 and b = 2, the following result will be obtained:
This equality is not true, since the number 3 is not equal to the number -3.